What reason do they have to believe this rather than “let’s blame the EU and make it look like reasonable regulations (being allowed to install app outside Apple’s control) are evil.”?

But since it is obvious the capability exists, they should allow other vendors the permissions needed to offer similar functionality with their own technology. These companies have been requesting this for years for their existing products, long before Apple introduced iPhone mirroring (long before even airplay).

iPhone mirroring is not innovation when you simply take your competitors product ideas and then block them from competing by giving yourself exclusive access to the very features they have been requesting for over 15 years...

Apple need not worry about being forced to "tweak the feature" if users are given the option to use an alternative solution on equal terms. Unless, of course, the aim is simply to stifle competition and keep users locked in through anti-consumer practices.